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a measurement of the loss of satisfaction in the case of a falling standard the expression

{(Q₁ − q₁)P₁ + (Q₂ − q₂)P₂ + . . .}/{Q₁P₁ + Q₂P₂ + . . .}

the ratio of the cost of the decrease in quantity to that of the quantities at the first date, both valued at the prices of the first date; this method leads to I₂ as the index of the increase of cost of living, but it is not of general application for it does not give equal importance to the distribution of expenditure at both dates since I₂ does not involve Q₁, Q₂ . . . .

T. L. Bennett (“Theory of Measurement of Changes in Cost of Living,” Journal of the Royal Statistical Society, May 1920) carries the argument further by important steps. With the notation already used, he supposes that a housekeeper gradually changes her purchases from quantities Q₁, Q₂ . . . at prices P₁, P₂ . . . to quantities q₁, q₂ . . . at prices p₁, p₂ . . ., the quantity of each commodity bought being related to its price by a law of demand. He then shows that the increase of expenditure, when the final is compared with the initial date. viz. e − E, is algebraically equal to X+L, where X = 1/2(Q₁ + q₁) (p₁ − P₁) + 1/2(Q₂ + q₂) (p₂ − P₂) + . . . and L = 1/2(q₁ − Q₁) (p₁ + P₁) + 1/2(q₂ − Q₂) (p₂ + P₂) + . . . and he identifies X as measuring the increased expenditure necessary to preserve the former standard of living and L as measuring increased satisfaction from increased consumption (or if L is negative a decrease).

This method gives a useful and simple test of the equivalence (as measured by satisfaction) of two budgets at different dates in the same country, for L should be zero, that is q₁⋅1/2(p₁ + P₁) + q₂⋅1/2(p₂ + P₂) + . . . should equal Q₁⋅1/2(p₁ + P₁) + Q₂⋅1/2(p₂ + P₂) + . . . . This test should be applied if the suggestion made above of constructing an equivalent budget for comparison is carried out.

If L is negative it would be necessary to add to expenditure e to make it equivalent to the earlier expenditure E, and Bennett, having regard to the changed purchasing power of money, suggests a somewhat complex and indefinite method of ascertaining the necessary amount; the index number for the cost of living may be written approximately 100 × (e+L^1/√100)/E, where I is chosen as one of the index numbers already written.

Case (d). The problem with which many countries were faced in 1920 and 1921 was in reality not that of preserving a standard of living on the level of 1914, but of adapting themselves to a lower average standard, whatever the fortunes of favoured classes. This may be illustrated by the arrangement of the salaries of civil servants in England in Feb. 1920. At that date the official measurement (on method b) of the increase in the cost of living over 1914 was 130%. The full increase of 130% was awarded to persons with a wage of 35s. weekly (£91 5s. per annum), 60% was added to the residue of salaries up to £200, and 45% to the residue of the salary. Thus a man whose salary was £400 on the pre-war basis received an addition of £273⅞ (130% on £91 5s. = £118⅝, 60% on £108 15s. = £65¼, 45% on £200 = £90), about 68% in all. This increment was increased or decreased by one twenty-sixth part for every complete movement of 5 points in the official index number averaged over certain periods. It appears to have been assumed on the one hand that the expenses of the middle class had not increased so much as indicated by the index number based on working-class expenditure, and on the other that the standard of living must be lowered—the higher the income the greater the fall. A similar scale was adopted at nearly the same date for railway officials. We are thus led to consider a conventional standard of living which changes from time to time. When there is no reference to a physiological minimum, the cost of living may be regarded as the cost of maintaining the standard customary to the social or occupational class concerned at a given time and place. In this sense the cost of living of Chinese labourers is lower than that of the Americans, though they pay the same prices for commodities. When “cost of living” is used in this sense it should always be accompanied by a reference to the standard attained. Thus the British Committee on the Cost of Living in 1918 estimated the average expenditure of working-class families in 1914 and 1918 and at the same time reported on the change of standard. In some of the statistics quoted below a conception of this kind is involved in the figure.

United Kingdom

(a) Cost of Food.—In the United Kingdom the basis of the official measurement of the cost of living is that of finding the cost of a standard budget of expenditure at various dates (see Labour Gazette, March 1920, p. 118, and Report on Working-Class Rents and Retail Prices, Cd.6955 of 1913, pp. 299 seq.). The standard budget was obtained from a collection of 1,944 records of weekly expenditure made in 1904; the average weekly family expenditure was 36s. 10d., of this 22s. 6d. was spent on food, and of the food 18s. 6d. is accounted for in the standard used prior to the war. A somewhat altered basis was taken in 1914. Rice, tapioca, oatmeal, pork, coffee, cocoa, jam, treacle, marmalade, currants and raisins (the expenditure on all of which was about 2s. 1d. in 1904) were omitted and fish and margarine added (an addition equivalent to 6d. in each case). It was assumed that, though prices had increased between 1904 and 1914, the relative expenditure (which alone enters into the computation) on the different commodities was unchanged; this assumption is too rigid but not unreasonable, and the facts otherwise known about price movements and consumption show that the error introduced is insignificant.

Relative importance being determined, the next step was to ascertain the movement of prices. Prior to 1914 the records were obtained exclusively for London, but it was shown (Cd.6955, pp. 299 and 306) that from 1907 to 1912 the average movement was very nearly the same in provincial towns as in London. From Aug. 1914 statements of prices were obtained for 650 towns and villages.

The index numbers of the cost of living, so far as food is concerned, were then obtained by the method b described above; prior to 1914, the year 1900 was taken as base and the prices then equated to 100; from the beginning of the war July 1914 was taken as base.

The index number is in the form 100 × (E₁r₁ + E₂r₂ + . . .) ÷ (E₁ + E₂ + . . . ) where E₁, E₂ . . . are the expenditures on the separate commodities in the standard budget and r₁, r₂ . . . are the ratios of the prices at any particular time to the prices at the basic date. The values actually taken for the E's were as in Table I. , being proportional to the expenditure.

Table I.

Bread	50
Flour	20
Rice[1]	3
Tapioca[1]	1
Oatmeal[1]	5
Potatoes	18

British meat:
Beef	24
Mutton	12
Pork[1]	15
Imported meat:
Beef	24
Mutton	12
Bacon	19
Fish[2]	9

Milk	25
Butter	41
Eggs	19
Cheese	10
Margarine[2]	10

Tea	22
Coffee[1]	2
Cocoa[1]	4

Sugar	19
Jam[1]	4
Treacle[1]	2
Marmalade[1]	4
Currants[1]	3
Raisins[1]	2

Totals prior to 1914	97
1914 and onwards	88

106 

95 

28 

34 

100 

105 

22 

19 

Grand totals: before 1914, 360; after, 334.

There are certain weaknesses in the method. It is assumed without explicit evidence that expenditure on meat was in the proportion 2s. on beef to 1s. on mutton, and that British and foreign meat were of equal importance, and the price ratios taken for meat are for four selected joints only; during the period 1915 to 1919, when the relative quantities available varied and relative prices were altered, this assumption affects the index numbers. The weight assigned to margarine is ar-

1. Omitted after 1914.
2. Omitted prior to 1914.